The fire broke out at Ninho do Urubu, a state-of-the-art training centre that was expanded and opened only two months ago after a £4.8m (R87m) investment. February 8, in an accommodation block at the facility of the club.
Television channel Globo reported that youth players were among the dead, although it did not cite its sources.
The fire followed days of heavy rains that hit the city and killed at least six people.
Flamengo, which launched the careers of World Cup victor Zico and Real Madrid phenom Vinicius Junior, houses many young players at its Ninho do Urubu facility.
The fire broke out at 5 am in the Vargem Grande district of Rio in a modern facility where the top-flight professional Flamengo squad also trains, the news website G1 said.
In addition to the deaths, three people were injured in the fire.
Debris from the fire at Flamengo football club's training complex litters the ground in Rio de Janeiro. "I just spoke to Caix Suarez and he is alive", said Rodrigues, adding that the youth told him he ran when he saw the flames in the morning.More news: Virginia Lt. Gov. Justin Fairfax accused of second sexual assault
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"Flamengo is in mourning" the team posted on its Twitter account.
Passion for soccer runs deep in Brazil, and as news of the tragedy spread, numerous world's best players and top officials expressed condolences.
The Ninho do Urubu, or Vulture's Nest is so called because the club's symbol is a vulture.
Flamengo won the Copa Libertadores in 1981 and won the Intercontinental Cup in the same year, beating Liverpool 3-0 in the final in Tokyo.
Brazil's Pele, widely regarded as the best player to ever live, said: "It's a very sad day for Brazilian football".
Zico, who is outside Brazil, wrote on Instagram: "What a shock getting this news here on the other side of the world", the former Brazil midfielder wrote.
Messages to Flamengo officials were not immediately answered.